{
  "id": "1709.03929",
  "version": "v1",
  "published": "2017-09-12T16:06:14.000Z",
  "updated": "2017-09-12T16:06:14.000Z",
  "title": "Simple modules over the Lie algebras of divergence zero vector fields on a torus",
  "authors": [
    "Brendan Frisk Dubsky",
    "Xianqian Guo",
    "Yufeng Yao",
    "Kaiming Zhao"
  ],
  "comment": "20 pages",
  "categories": [
    "math.RT",
    "math.QA",
    "math.RA"
  ],
  "abstract": "Let $n\\ge2$ be an integer, $\\mathcal{K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\\mathbb{C} [x_1^{\\pm1}, x_2^{\\pm1}, ..., x_n^{\\pm1}]$, and $\\mathbb{S}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. For any $\\mathfrak{sl}_n$-module $V$ and any module $P$ over $\\mathcal{K}_n$, we define an $\\mathbb{S}_n$-module structure on the tensor product $P\\otimes V$. In this paper, necessary and sufficient conditions for the $\\mathbb{S}_n$-modules $P\\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule $\\mathbb{S}_n$-modules is provided. More precisely, all nonminuscule $\\mathbb{S}_n$-modules are simple, and pairwise nonisomorphic. For minuscule $\\mathbb{S}_n$-modules, minimal and maximal submodules are concretely constructed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T16:06:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B65",
      "17B66"
    ],
    "keywords": [
      "divergence zero vector fields",
      "lie algebra",
      "simple modules",
      "laurent polynomial algebra",
      "module structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}